Compound Interest

Percentage Accurate: 28.9% → 94.3%

Time: 14.5s

Alternatives: 13

Speedup: 8.1×

• could not determine a ground truth

  1. i = 1.2749511841393868e-290
  2. n = -2.8101834613240654e+39

Specification

Math

\[\begin{array}{l} \\ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 28.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-262}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot t\_0 - n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, n \cdot 100, 0\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 5e-262)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_1 INFINITY)
       (* (/ 100.0 i) (- (* n t_0) n))
       (/ (fma i (* n 100.0) 0.0) i)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 5e-262) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (100.0 / i) * ((n * t_0) - n);
	} else {
		tmp = fma(i, (n * 100.0), 0.0) / i;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 5e-262)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(100.0 / i) * Float64(Float64(n * t_0) - n));
	else
		tmp = Float64(fma(i, Float64(n * 100.0), 0.0) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-262], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(100.0 / i), $MachinePrecision] * N[(N[(n * t$95$0), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(n * 100.0), $MachinePrecision] + 0.0), $MachinePrecision] / i), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 4.99999999999999992e-262

   1. Initial program 27.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      2. lift-pow.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{\frac{i}{n}} \]

      3. pow-to-exp N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \]

      4. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

      5. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      6. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      7. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \color{blue}{\left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      8. lower-log1p.f64 98.1

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   4. Applied rewrites 98.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   if 4.99999999999999992e-262 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 97.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      2. lift-pow.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{\frac{i}{n}} \]

      3. pow-to-exp N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \]

      4. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

      5. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      6. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      7. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \color{blue}{\left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      8. lower-log1p.f64 44.4

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   4. Applied rewrites 44.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}} \]

      4. lift-expm1.f64 N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} - 1\right)}}{\frac{i}{n}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}} - 1\right)}{\frac{i}{n}} \]

      6. *-commutative N/A

         \[\leadsto \frac{100 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1\right)}{\frac{i}{n}} \]

      7. lift-log1p.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n} - 1\right)}{\frac{i}{n}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\log \color{blue}{\left(1 + \frac{i}{n}\right)} \cdot n} - 1\right)}{\frac{i}{n}} \]

      9. pow-to-exp N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)}{\frac{i}{n}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{100 \cdot \left({\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1\right)}{\frac{i}{n}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{100 \cdot \left({\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1\right)}{\frac{i}{n}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{\frac{i}{n}}} \]

      13. div-inv N/A

          \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{i \cdot \frac{1}{n}}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i \cdot \color{blue}{\frac{1}{n}}} \]

      15. times-frac N/A

          \[\leadsto \color{blue}{\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}} \]

   6. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{100}{i} \cdot \left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right)} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 1}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 1}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 1}{\color{blue}{\frac{1}{n} \cdot i}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      11. times-frac N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}} \cdot \frac{1}{i}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \frac{1}{i}, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}}, \frac{1}{i}, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{1}{n}}}, \frac{1}{i}, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \color{blue}{\frac{1}{i}}, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \frac{1}{i}, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \frac{1}{i}, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      18. lower-neg.f64 3.7

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \frac{1}{i}, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 3.7%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \frac{1}{i}, \frac{-n}{i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{100 \cdot \left(i \cdot n\right) + 100 \cdot \left(n + -1 \cdot n\right)}{i}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{100 \cdot \left(n + -1 \cdot n\right) + 100 \cdot \left(i \cdot n\right)}}{i} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\left(100 \cdot n + 100 \cdot \left(-1 \cdot n\right)\right)} + 100 \cdot \left(i \cdot n\right)}{i} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{100 \cdot \left(n + -1 \cdot n\right)} + 100 \cdot \left(i \cdot n\right)}{i} \]

      4. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)} + 100 \cdot \left(i \cdot n\right)}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right) + 100 \cdot \left(i \cdot n\right)}{i} \]

      6. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0} + 100 \cdot \left(i \cdot n\right)}{i} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} + 100 \cdot \left(i \cdot n\right)}{i} \]

      8. *-commutative N/A

         \[\leadsto \frac{0 + \color{blue}{\left(i \cdot n\right) \cdot 100}}{i} \]

      9. associate-*r* N/A

         \[\leadsto \frac{0 + \color{blue}{i \cdot \left(n \cdot 100\right)}}{i} \]

      10. *-commutative N/A

          \[\leadsto \frac{0 + i \cdot \color{blue}{\left(100 \cdot n\right)}}{i} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{0 + i \cdot \left(100 \cdot n\right)}{i}} \]

   7. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(i, n \cdot 100, 0\right)}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-262}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, n \cdot 100, 0\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-262}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot t\_0 - n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, n \cdot 100, 0\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 5e-262)
     (* (* n 100.0) (/ (expm1 (* n (log1p (/ i n)))) i))
     (if (<= t_1 INFINITY)
       (* (/ 100.0 i) (- (* n t_0) n))
       (/ (fma i (* n 100.0) 0.0) i)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 5e-262) {
		tmp = (n * 100.0) * (expm1((n * log1p((i / n)))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (100.0 / i) * ((n * t_0) - n);
	} else {
		tmp = fma(i, (n * 100.0), 0.0) / i;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 5e-262)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(100.0 / i) * Float64(Float64(n * t_0) - n));
	else
		tmp = Float64(fma(i, Float64(n * 100.0), 0.0) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-262], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(100.0 / i), $MachinePrecision] * N[(N[(n * t$95$0), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(n * 100.0), $MachinePrecision] + 0.0), $MachinePrecision] / i), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 4.99999999999999992e-262

   1. Initial program 27.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \cdot 100 \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{\frac{i}{n}}} \cdot 100 \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}} \cdot \left(n \cdot 100\right) \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \left(n \cdot 100\right) \]

      10. sub-neg N/A

          \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)}}{i} \cdot \left(n \cdot 100\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)}}{i} \cdot \left(n \cdot 100\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

      13. lower-*.f64 27.5

          \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \color{blue}{\left(n \cdot 100\right)} \]

   4. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}{i} \cdot \left(n \cdot 100\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} + -1}{i} \cdot \left(n \cdot 100\right) \]

      3. pow-to-exp N/A

         \[\leadsto \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} + -1}{i} \cdot \left(n \cdot 100\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{e^{\log \color{blue}{\left(1 + \frac{i}{n}\right)} \cdot n} + -1}{i} \cdot \left(n \cdot 100\right) \]

      5. lift-log1p.f64 N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n} + -1}{i} \cdot \left(n \cdot 100\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}} + -1}{i} \cdot \left(n \cdot 100\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}} + -1}{i} \cdot \left(n \cdot 100\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{i} \cdot \left(n \cdot 100\right) \]

      9. sub-neg N/A

         \[\leadsto \frac{\color{blue}{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} - 1}}{i} \cdot \left(n \cdot 100\right) \]

      10. lift-expm1.f64 96.7

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{i} \cdot \left(n \cdot 100\right) \]

   6. Applied rewrites 96.7%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{i} \cdot \left(n \cdot 100\right) \]

   if 4.99999999999999992e-262 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 98.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      2. lift-pow.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{\frac{i}{n}} \]

      3. pow-to-exp N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \]

      4. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

      5. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      6. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      7. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \color{blue}{\left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      8. lower-log1p.f64 61.7

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   4. Applied rewrites 61.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}} \]

      4. lift-expm1.f64 N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} - 1\right)}}{\frac{i}{n}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}} - 1\right)}{\frac{i}{n}} \]

      6. *-commutative N/A

         \[\leadsto \frac{100 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1\right)}{\frac{i}{n}} \]

      7. lift-log1p.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n} - 1\right)}{\frac{i}{n}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\log \color{blue}{\left(1 + \frac{i}{n}\right)} \cdot n} - 1\right)}{\frac{i}{n}} \]

      9. pow-to-exp N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)}{\frac{i}{n}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{100 \cdot \left({\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1\right)}{\frac{i}{n}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{100 \cdot \left({\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1\right)}{\frac{i}{n}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{\frac{i}{n}}} \]

      13. div-inv N/A

          \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{i \cdot \frac{1}{n}}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i \cdot \color{blue}{\frac{1}{n}}} \]

      15. times-frac N/A

          \[\leadsto \color{blue}{\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}} \]

   6. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{100}{i} \cdot \left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right)} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 1}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 1}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 1}{\color{blue}{\frac{1}{n} \cdot i}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      11. times-frac N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}} \cdot \frac{1}{i}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \frac{1}{i}, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}}, \frac{1}{i}, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{1}{n}}}, \frac{1}{i}, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \color{blue}{\frac{1}{i}}, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \frac{1}{i}, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \frac{1}{i}, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      18. lower-neg.f64 3.3

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \frac{1}{i}, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 3.3%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}}, \frac{1}{i}, \frac{-n}{i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{100 \cdot \left(i \cdot n\right) + 100 \cdot \left(n + -1 \cdot n\right)}{i}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{100 \cdot \left(n + -1 \cdot n\right) + 100 \cdot \left(i \cdot n\right)}}{i} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\left(100 \cdot n + 100 \cdot \left(-1 \cdot n\right)\right)} + 100 \cdot \left(i \cdot n\right)}{i} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{100 \cdot \left(n + -1 \cdot n\right)} + 100 \cdot \left(i \cdot n\right)}{i} \]

      4. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)} + 100 \cdot \left(i \cdot n\right)}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right) + 100 \cdot \left(i \cdot n\right)}{i} \]

      6. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0} + 100 \cdot \left(i \cdot n\right)}{i} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} + 100 \cdot \left(i \cdot n\right)}{i} \]

      8. *-commutative N/A

         \[\leadsto \frac{0 + \color{blue}{\left(i \cdot n\right) \cdot 100}}{i} \]

      9. associate-*r* N/A

         \[\leadsto \frac{0 + \color{blue}{i \cdot \left(n \cdot 100\right)}}{i} \]

      10. *-commutative N/A

          \[\leadsto \frac{0 + i \cdot \color{blue}{\left(100 \cdot n\right)}}{i} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{0 + i \cdot \left(100 \cdot n\right)}{i}} \]

   7. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(i, n \cdot 100, 0\right)}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-262}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, n \cdot 100, 0\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 34.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024228 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))